Summary: The paper deals with the vertical dynamics of a railway track. In the system under consideration a single rail is modelled as a Timoshenko beam. The rails are coupled by means of a number of periodically spaced sleepers which are modelled as rigid bodies with two degrees of freedom. The case of free wave propagation is investigated in detail. The method used consists in the application of Floquet’s theorem to the differential equation of motion of the beam. There are two forms of travelling wave in the case of an unloaded two-dimensional periodic structure. The first form corresponds to the in-phase propagation of waves in the two rails, the second form represents the case of a half-wave-length phase difference between the propagating waves. The method to obtain the solution for the system under moving harmonic forces is briefly discussed.